I’ll echo the recommendation of Neon Genesis Evangelion, particularly if you’re looking for a story with insight into the psychological condition and experiences of the protagonist.

If you’re in the mood to ponder other ideas philosophically, you may want to try Time of Eve (Eve no Jikan), Plastic Memories, and Beatless. All of them are stories involving androids, but they explore different concepts. Time of Eve explores issues of identity and presentation (in some ways that are surprisingly salient today), Plastic Memories explores end-of-life considerations, and Beatless explores the roles of AI in the future of human society.

For a couple series with dystopian futures, consider Coppelion and Planetarian. Coppelion is about people who venture into the highly radioactive ruins of Tokyo after a nuclear accident; its visual style deliberately makes use of washed-out colors. Planetarian is about a scrap collector who ventures into a ruined city only to encounter an android who seems oblivious of the ruins around her; her wordy and oblivious optimism contrasts sharply with his cynicism, in this relatively short and also somewhat philosophical anime series. (It’s also available in the form of a visual novel.)

For something more action-packed, there’s Guilty Crown and the Symphogear series. Guilty Crown seems to have some polarizing opinions, though not for lack of effort in the production – it’s quite a spectacle, visually and musically, and it was definitely meant to go big. On one hand I felt disappointed by it but on the other hand I also feel like I really wanted to like it. As with any show, though, you should be forming your opinion of things anyway.

For drama, I’d suggest The Price of Smiles (Egao no Daika) and Str.A.In: Strategic Armored Infantry (Soukou no Strain). Both of them revolve around war and associated tragedies, but in settings with futuristic combat mechs and other technological advancements. The latter takes place in space.

For something with a brighter but still futuristic vibe, try Stellvia of the Universe (Uchuu no Stellvia). Here’s a story about high school in space, with all the laughs, goofs, friendship, smiles, and even romance, as well as interpersonal drama and expectations – particularly heavy expectations at that. As a point of trivia, it’s pretty neat that the main character specializes in programming.

Edit: I wanted to throw in one more recommendation, the Nanoha series. This is a magical girl series that’s renowned for incorporating Gundam-inspired magitech elements, particularly in the various magical abilities the characters use. It’s also got both drama and positive vibes. Good for an affirmation of the power of friendship, backed up by a big stick!

Things that come to mind quickly are Ghost in the Shell, Ergo Proxy, Texhnolyze, Cyberpunk Edgerunners, Psycho-pass.

Serial Experiments Lain isn’t really cyberpunk but has aged to be retrofuturism.

Seconding Akira and Ghost in the shell, Steins;Gate is great too.

One of the first ones I saw (and I love cyberpunk and scifi shit) was Serial Experiments Lain and it was one of the first ones to get me absolutely hooked. Mind you, it is weird and strange, but if you’re into stuff like David Lynch at all you’ll love it.

Others to look into that aren’t exactly cyberpunk, but futuristic/scifi nonetheless: Neon Genesis Evangelion (finish it, it gets really cool towards the end and I thought all things mecha-anything was just dudes being silly about robots), Cowboy Bebop, Ergo Proxy and Gantz (might seem silly at first, but gets interesting towards the end).

Psycho-Pass might be up your alley. It’s about Japanese society under the Sibyl System, a system that determines how likely you are to do crimes. Depending on your score, you could get jailed or killed by police.